65配2 Budynas-Nisbett:Shigley's I Design of Mechanical 13.Gears-General T©The McGraw-Hfll Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 13 Gears-General Chapter Outline 13-1 Types of Gears 654 13-2 Nomenclature 655 13-3 Conjugate Action 657 13-4 Involute Properties 658 13-5 Fundamentals 658 13-6 Contact Ratio 664 13-7 Interference 665 13-8 The Forming of Gear Teeth 667 13-9 Straight Bevel Gears 670 13-10 Parallel Helical Gears 671 13-11 Worm Gears 675 13-12 Tooth Systems 676 13-13 Gear Trains 678 13-14 Force Analysis-Spur Gearing 685 13-15 Force Analysis-Bevel Gearing 689 13-16 Force Analysis-Helical Gearing 692 13-17 Force Analysis-Worm Gearing 694 653
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General 652 © The McGraw−Hill Companies, 2008 Chapter Outline 13–1 Types of Gears 654 13–2 Nomenclature 655 13–3 Conjugate Action 657 13–4 Involute Properties 658 13–5 Fundamentals 658 13–6 Contact Ratio 664 13–7 Interference 665 13–8 The Forming of Gear Teeth 667 13–9 Straight Bevel Gears 670 13–10 Parallel Helical Gears 671 13–11 Worm Gears 675 13–12 Tooth Systems 676 13–13 Gear Trains 678 13–14 Force Analysis—Spur Gearing 685 13–15 Force Analysis—Bevel Gearing 689 13–16 Force Analysis—Helical Gearing 692 13–17 Force Analysis—Worm Gearing 694 13Gears—General 653
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 13.Gears-General T©The McGraw-Hill 653 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 654 Mechanical Engineering Design This chapter addresses gear geometry,the kinematic relations,and the forces transmit- ted by the four principal types of gears:spur,helical,bevel,and worm gears.The forces transmitted between meshing gears supply torsional moments to shafts for motion and power transmission and create forces and moments that affect the shaft and its bearings. The next two chapters will address stress,strength,safety,and reliability of the four types of gears. 13-1 Types of Gears Spur gears,illustrated in Fig.13-1,have teeth parallel to the axis of rotation and are used to transmit motion from one shaft to another,parallel,shaft.Of all types,the spur gear is the simplest and,for this reason,will be used to develop the primary kinematic relationships of the tooth form. Helical gears,shown in Fig.13-2,have teeth inclined to the axis of rotation.Helical gears can be used for the same applications as spur gears and,when so used,are not as noisy,because of the more gradual engagement of the teeth during meshing.The inclined tooth also develops thrust loads and bending couples.which are not present with spur gearing.Sometimes helical gears are used to transmit motion between nonparallel shafts. Bevel gears,shown in Fig.13-3,have teeth formed on conical surfaces and are used mostly for transmitting motion between intersecting shafts.The figure actually illustrates straight-tooth bevel gears.Spiral bevel gears are cut so the tooth is no longer straight,but forms a circular arc.Hypoid gears are quite similar to spiral bevel gears except that the shafts are offset and nonintersecting. Figure 13-1 Spur gears are used to transmit rotary motion between parallel shafts. IⅧ Figure 13-2 Helical gears are used to transmit motion between parallel or nonparallel shafts
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General © The McGraw−Hill 653 Companies, 2008 654 Mechanical Engineering Design Figure 13–1 Spur gears are used to transmit rotary motion between parallel shafts. Figure 13–2 Helical gears are used to transmit motion between parallel or nonparallel shafts. This chapter addresses gear geometry, the kinematic relations, and the forces transmitted by the four principal types of gears: spur, helical, bevel, and worm gears. The forces transmitted between meshing gears supply torsional moments to shafts for motion and power transmission and create forces and moments that affect the shaft and its bearings. The next two chapters will address stress, strength, safety, and reliability of the four types of gears. 13–1 Types of Gears Spur gears, illustrated in Fig. 13–1, have teeth parallel to the axis of rotation and are used to transmit motion from one shaft to another, parallel, shaft. Of all types, the spur gear is the simplest and, for this reason, will be used to develop the primary kinematic relationships of the tooth form. Helical gears,shown in Fig. 13–2, have teeth inclined to the axis of rotation. Helical gears can be used for the same applications as spur gears and, when so used, are not as noisy, because of the more gradual engagement of the teeth during meshing. The inclined tooth also develops thrust loads and bending couples, which are not present with spur gearing. Sometimes helical gears are used to transmit motion between nonparallel shafts. Bevel gears, shown in Fig. 13–3, have teeth formed on conical surfaces and are used mostly for transmitting motion between intersecting shafts. The figure actually illustrates straight-tooth bevel gears. Spiral bevel gears are cut so the tooth is no longer straight, but forms a circular arc. Hypoid gears are quite similar to spiral bevel gears except that the shafts are offset and nonintersecting
654 Budynas-Nisbett:Shigley's Ill.Design of Mechanical 13.Gears-General T©The McGraw-Hil Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Gears-General 655 Figure 13-3 Bevel gears are used to transmit rotary motion between intersecting shafts. Figure 13-4 Worm gearsets are used to transmit rotary motion between nonparallel and nonintersecting shafts. Worms and worm gears,shown in Fig.13-4,represent the fourth basic gear type. As shown,the worm resembles a screw.The direction of rotation of the worm gear,also called the worm wheel,depends upon the direction of rotation of the worm and upon whether the worm teeth are cut right-hand or left-hand.Worm-gear sets are also made so that the teeth of one or both wrap partly around the other.Such sets are called single- enveloping and double-enveloping worm-gear sets.Worm-gear sets are mostly used when the speed ratios of the two shafts are quite high,say,3 or more. 13-2 Nomenclature The terminology of spur-gear teeth is illustrated in Fig.13-5.The pitch circle is a the- oretical circle upon which all calculations are usually based;its diameter is the pitch diameter.The pitch circles of a pair of mating gears are tangent to each other.A pinion is the smaller of two mating gears.The larger is often called the gear. The circular pitch p is the distance,measured on the pitch circle,from a point on one tooth to a corresponding point on an adjacent tooth.Thus the circular pitch is equal to the sum of the tooth thickness and the width of space
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General 654 © The McGraw−Hill Companies, 2008 Gears—General 655 Figure 13–3 Bevel gears are used to transmit rotary motion between intersecting shafts. Worms and worm gears, shown in Fig. 13–4, represent the fourth basic gear type. As shown, the worm resembles a screw. The direction of rotation of the worm gear, also called the worm wheel, depends upon the direction of rotation of the worm and upon whether the worm teeth are cut right-hand or left-hand. Worm-gear sets are also made so that the teeth of one or both wrap partly around the other. Such sets are called singleenveloping and double-enveloping worm-gear sets. Worm-gear sets are mostly used when the speed ratios of the two shafts are quite high, say, 3 or more. 13–2 Nomenclature The terminology of spur-gear teeth is illustrated in Fig. 13–5. The pitch circle is a theoretical circle upon which all calculations are usually based; its diameter is the pitch diameter. The pitch circles of a pair of mating gears are tangent to each other. A pinion is the smaller of two mating gears. The larger is often called the gear. The circular pitch p is the distance, measured on the pitch circle, from a point on one tooth to a corresponding point on an adjacent tooth. Thus the circular pitch is equal to the sum of the tooth thickness and the width of space. Figure 13–4 Worm gearsets are used to transmit rotary motion between nonparallel and nonintersecting shafts
Budynas-Nisbett:Shigley's Ill.Design of Mechanical 13.Gears-General T©The McGraw-Hill 65 Mechanical Engineering Elements Companies,2008 Design,Eighth Edition 656 Mechanical Engineering Design Figure 13-5 Top land Nomenclature of spurgear teeth. Face wid由h Addendum circle Circular pitch Flank Tooth Pitch circle thickness Width of space Bottom land Clearance Fillet radius Dedendum Clearance circle circle The module m is the ratio of the pitch diameter to the number of teeth.The cus- tomary unit of length used is the millimeter.The module is the index of tooth size in SI. The diametral pitch P is the ratio of the number of teeth on the gear to the pitch diameter.Thus,it is the reciprocal of the module.Since diametral pitch is used only with U.S.units,it is expressed as teeth per inch. The addendum a is the radial distance between the top land and the pitch circle. The dedendum b is the radial distance from the bottom land to the pitch circle.The whole depth h,is the sum of the addendum and the dedendum. The clearance circle is a circle that is tangent to the addendum circle of the mat- ing gear.The clearance c is the amount by which the dedendum in a given gear exceeds the addendum of its mating gear.The backlash is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth measured on the pitch circles. You should prove for yourself the validity of the following useful relations: P= d (13-1) d m二 (13-21 πd P=N =πm (13-3) pP=π (13-4) where P=diametral pitch,teeth per inch N=number of teeth d=pitch diameter,in m=module,mm d=pitch diameter,mm p circular pitch
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General © The McGraw−Hill 655 Companies, 2008 656 Mechanical Engineering Design Addendum Dedendum Clearance Bottom land Fillet radius Dedendum circle Clearance circle Tooth thickness Face width Width of space Face Top land Addendum circle Pitch circle Flank Circular pitch Figure 13–5 Nomenclature of spur-gear teeth. The module m is the ratio of the pitch diameter to the number of teeth. The customary unit of length used is the millimeter. The module is the index of tooth size in SI. The diametral pitch P is the ratio of the number of teeth on the gear to the pitch diameter. Thus, it is the reciprocal of the module. Since diametral pitch is used only with U.S. units, it is expressed as teeth per inch. The addendum a is the radial distance between the top land and the pitch circle. The dedendum b is the radial distance from the bottom land to the pitch circle. The whole depth ht is the sum of the addendum and the dedendum. The clearance circle is a circle that is tangent to the addendum circle of the mating gear. The clearance c is the amount by which the dedendum in a given gear exceeds the addendum of its mating gear. The backlash is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth measured on the pitch circles. You should prove for yourself the validity of the following useful relations: P = N d (13–1) m = d N (13–2) p = πd N = πm (13–3) pP = π (13–4) where P = diametral pitch, teeth per inch N = number of teeth d = pitch diameter, in m = module, mm d = pitch diameter, mm p = circular pitch
656 Budynas-Nisbett Shigley's IIl Design of Mechanical 13.Gears-General T©The McGraw-Hil Mechanical Engineering Elements Companies,2008 Design,Eighth Edition Gears-General 657 13-3 Conjugate Action The following discussion assumes the teeth to be perfectly formed,perfectly smooth, and absolutely rigid.Such an assumption is,of course,unrealistic,because the applica- tion of forces will cause deflections. Mating gear teeth acting against each other to produce rotary motion are similar to cams.When the tooth profiles,or cams,are designed so as to produce a constant angular- velocity ratio during meshing,these are said to have conjugate action.In theory,at least, it is possible arbitrarily to select any profile for one tooth and then to find a profile for the meshing tooth that will give conjugate action.One of these solutions is the imolute profile, which,with few exceptions,is in universal use for gear teeth and is the only one with which we should be concerned. When one curved surface pushes against another(Fig.13-6),the point of contact occurs where the two surfaces are tangent to each other(point c),and the forces at any instant are directed along the common normal ab to the two curves.The line ab,rep- resenting the direction of action of the forces,is called the line of action.The line of action will intersect the line of centers O-0 at some point P.The angular-velocity ratio between the two arms is inversely proportional to their radii to the point P.Circles drawn through point P from each center are called pitch circles,and the radius of each circle is called the pitch radius.Point P is called the pitch point. Figure 13-6 is useful in making another observation.A pair of gears is really pairs of cams that act through a small arc and,before running off the involute contour, are replaced by another identical pair of cams.The cams can run in either direction and are configured to transmit a constant angular-velocity ratio.If involute curves are used, the gears tolerate changes in center-to-center distance with no variation in constant angular-velocity ratio.Furthermore,the rack profiles are straight-fanked,making pri- mary tooling simpler. To transmit motion at a constant angular-velocity ratio,the pitch point must remain fixed;that is,all the lines of action for every instantaneous point of contact must pass through the same point P.In the case of the involute profile,it will be shown that all points of contact occur on the same straight line ab,that all normals to the tooth profiles at the point of contact coincide with the line ab,and,thus,that these profiles transmit uniform rotary motion. Figure 13-6 Cam A and follower B in contact.When the contacting surfaces are involute profiles the ensuing conjugate action produces a constant angularvelocity ratio
Budynas−Nisbett: Shigley’s Mechanical Engineering Design, Eighth Edition III. Design of Mechanical Elements 13. Gears — General 656 © The McGraw−Hill Companies, 2008 Gears—General 657 O B rB rA b c a A O P Figure 13–6 Cam A and follower B in contact. When the contacting surfaces are involute profiles, the ensuing conjugate action produces a constant angular-velocity ratio. 13–3 Conjugate Action The following discussion assumes the teeth to be perfectly formed, perfectly smooth, and absolutely rigid. Such an assumption is, of course, unrealistic, because the application of forces will cause deflections. Mating gear teeth acting against each other to produce rotary motion are similar to cams. When the tooth profiles, or cams, are designed so as to produce a constant angularvelocity ratio during meshing, these are said to have conjugate action. In theory, at least, it is possible arbitrarily to select any profile for one tooth and then to find a profile for the meshing tooth that will give conjugate action. One of these solutions is the involute profile, which, with few exceptions, is in universal use for gear teeth and is the only one with which we should be concerned. When one curved surface pushes against another (Fig. 13–6), the point of contact occurs where the two surfaces are tangent to each other (point c), and the forces at any instant are directed along the common normal ab to the two curves. The line ab, representing the direction of action of the forces, is called the line of action. The line of action will intersect the line of centers O-O at some point P. The angular-velocity ratio between the two arms is inversely proportional to their radii to the point P. Circles drawn through point P from each center are called pitch circles, and the radius of each circle is called the pitch radius. Point P is called the pitch point. Figure 13–6 is useful in making another observation. A pair of gears is really pairs of cams that act through a small arc and, before running off the involute contour, are replaced by another identical pair of cams. The cams can run in either direction and are configured to transmit a constant angular-velocity ratio. If involute curves are used, the gears tolerate changes in center-to-center distance with no variation in constant angular-velocity ratio. Furthermore, the rack profiles are straight-flanked, making primary tooling simpler. To transmit motion at a constant angular-velocity ratio, the pitch point must remain fixed; that is, all the lines of action for every instantaneous point of contact must pass through the same point P. In the case of the involute profile, it will be shown that all points of contact occur on the same straight line ab, that all normals to the tooth profiles at the point of contact coincide with the line ab, and, thus, that these profiles transmit uniform rotary motion